Scattering diagrams and stability conditions
Tom Sutherland (Pavia)
Aula Beltrami - Thursday, April 28, 2016 h.17:00
Abstract. Motivated by considerations in mirror symmetry, to a toric degeneration of a Calabi-Yau variety one can associate a scattering diagram. By work of Gross-Siebert the original variety can be reconstructed from the scattering diagram along with a canonical basis of functions, generalizing the theta functions associated to the Mumford degeneration of a toric variety which have an "empty" scattering diagram. These ideas have been subsequently applied to a certain class of open Calabi-Yau varieties called cluster varieties possessing toric degenerations to an algebraic torus labelled by so-called clusters.
Following a recent article of Bridgeland I will show how to construct a scattering diagram on a space of stability conditions of a Calabi-Yau-3 category associated to each heart of a bounded t-structure. For a certain class of cluster varieties, there exists a corresponding Calabi-Yau-3 triangulated category whose hearts "categorify" their clusters, and for which Bridgeland's construction of a scattering diagram associated to a heart agrees to the Gross-Siebert scattering diagram associated to a cluster. This allows us to study the geometry of these cluster varieties through enumerative invariants of the category which "count" semistable objects of the category.
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